All of the 3rd grade teachers and students from Loyola went on a field trip to an archaeology museum. Tickets were $$8.00$ each for teachers and $$4.50$ each for students, and the group paid $$38.50$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$16.00$ each for teachers and $$8.50$ each for students, and the group paid $$74.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+4.5y = 38.5}$ ${16x+8.5y = 74.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-16x-9y = -77}$ ${16x+8.5y = 74.5}$ Add the top and bottom equations together. $ -0.5y = -2.5 $ $ y = \dfrac{-2.5}{-0.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {8x+4.5y = 38.5}$ to find $x$ ${8x + 4.5}{(5)}{= 38.5}$ $8x+22.5 = 38.5$ $8x = 16$ $x = \dfrac{16}{8}$ ${x = 2}$ You can also plug ${y = 5}$ into $ {16x+8.5y = 74.5}$ and get the same answer for $x$ ${16x + 8.5}{(5)}{= 74.5}$ ${x = 2}$ There were $2$ teachers and $5$ students on the field trips.